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The higher you stand above the moon, the less gravitational pull it has. And there is no air resistance to dampen the acceleration. So if you keep dropping objects at higher and higher distances from the lunar surface, I have this idea that the speed of the object dropped when it hits the ground will converge to a finite value. I'm just curious about what this finite value would be. (Neglecting the earth's presence and all that.) Now of course you could THROW the object to speed it up, but that's not what I'm asking here. Gravitational force only.

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    $\begingroup$ yes, the maximum would be the escape velocity, the one it will have if you drop it from infinitely far away .en.wikipedia.org/wiki/Escape_velocity $\endgroup$ – user126422 Jan 31 '17 at 0:50
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Since there is no drag force you can just use mechanical energy conservation, that is, $$\Delta E=\Delta K+\Delta U=0.$$

If the object of mass $m$ is released with initial velocity $v_0$ and initial height (above surface) $x_0$, then the above equation reads $$\frac 12m(v^2-v_0^2)=GMm\left(\frac{1}{R}-\frac{1}{x_0} \right),$$ where $M$ and $R$ are the Moon's mass and radius, respectively and $G$ the gravitation constant. Solving for $v$ you get $$v=\sqrt{v_{i}^2+2GM\left(\frac{1}{R}-\frac{1}{x_0} \right)}.$$

The largest velocity possible at the surface requires that you drop the mass from infinity which correspond to take the limit $x_0\rightarrow\infty$. If you release it from rest then this velocity matches the escape velocity. This was expected since there is no energy dissipation, the up and down trajectories must be completely symmetric.

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When the body (mass $m$) reaches the moon surface from infinity the potential energy of the body will be converted into kinetic energy. At the moon surface its potential energy is $-G\frac{mM}{R}$ with $M$ the mass and $R$ the radius of the moon. Setting that equal to the kinetic energy gives $v = \sqrt{2GM/R}$. This is also called the escape velocity i.e. the velocity it needs to be given to escape the moon's gravitation field.

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